On intersections of conjugacy classes and Bruhat cells

2009/06/12 by Kei Yuen Chan, Jiang-Hua Lu, Chan, Kei Yuen +3
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.0906.2254

20 pages. Revised version. to appear in Transformation Groups

arxiv created 2010/01/21 · arxiv updated 2010/02/08

Abstract

For a connected complex semi-simple Lie group G and a fixed pair (B, B^-) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB^- is non-empty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution \mc ∈ W associated to C. We study properties of the elements \mc. For G = SL(n+1, \Cset), we describe \mc explicitly for every conjugacy class C, and for the case when w ∈ W is an involution, we also give an explicit answer to when C ∩ (BwB) is non-empty.

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